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Trigonometry

Trigonometry — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionWorking with Two Sides and Included Angle SASConcept Practice
2 marks~3 minCriterion A
A surveyor measures two boundary lines from a fixed point AA: line AB=5 cmAB = 5 \text{ cm} and line AC=7 cmAC = 7 \text{ cm}, with ∠BAC=40°\angle BAC = 40°.

Calculate the length of BCBC, giving your answer correct to 3 significant figures. [2]
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2QuestionMulti-Step Navigation ProblemsConcept Practice
2 marks~3 minCriterion A
A coast guard vessel is at point PP and a lighthouse is at point QQ. The diagram shows a north arrow at PP and the line PQPQ.

The angle measured clockwise from north to line PQPQ is 47°47°.
a
Deduce the three-figure bearing of QQ from PP. [1]
b
The coast guard must sail on a bearing between 040°040° and 055°055° to stay within a safe navigation corridor. Justify whether the vessel is heading within the safe corridor. [1]
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3QuestionUsing SOH-CAH-TOA to Find Missing SidesConcept Practice
4 marks~6 minCriterion A
A solar panel installation company assesses whether a roof section is suitable for fitting panels. The slant length of the roof is 15 m and the angle between the slant and the horizontal is 40°.
a
Write down the trigonometric ratio that links the adjacent side, the hypotenuse, and the angle in a right triangle. [1]
b
Calculate the horizontal length of the roof section. [2]
c
The installer states: "A horizontal span greater than 11 m confirms the roof is wide enough for the panel." Advise the installer whether the roof section should be approved for panel installation, justifying your answer with reference to your result from part (b). [1]
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4QuestionSketching Graphs of y = sin x cos x tan xConcept Practice
2 marks~3 minCriterion A
A buoy bobs in the ocean. Its vertical displacement, dd metres, from its rest position is modelled by d=sin⁡xd = \sin x, where xx is measured in degrees and 0°≤x≤360°0° \leq x \leq 360° represents one tidal cycle.
a
Construct a sketch of y=sin⁡xy = \sin x for −360°≤x≤360°-360° \leq x \leq 360°, plotting key points at multiples of 90°90° and drawing a smooth curve through them. [1]
b
The buoy is considered "active" when d>0d > 0. Interpret how many degrees of each 360°360° cycle the buoy spends above its rest position, and explain what this reveals about the symmetry of the model. [1]
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5QuestionSin and Cos rules based problemsConcept Practice
2 marks~3 minCriterion A
A surveyor measures a triangular plot of land, ABCABC. Angle A=35°A = 35°, angle B=72°B = 72°, and side a=8.2a = 8.2 cm (opposite angle AA).

Calculate the length of side bb, opposite angle BB, using the sine rule:

asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

Give your answer in centimetres, correct to 3 significant figures. [2]
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6QuestionProblem Solving with Non-Right TrianglesConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a triangular garden plot. The plot has two boundary sides of length AB=7AB = 7 m and BC=9BC = 9 m, with the angle between them measuring ∠ABC=40°\angle ABC = 40°.
a
Deduce which angle must be substituted into the formula Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C when a=7a = 7 m and b=9b = 9 m, and justify your reasoning. [1]
b
Calculate the area of the triangular plot. [1]
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7QuestionWorking with Two Sides and Included Angle SASAssessment Practice
4 marks~6 minCriterion B
A landscape architect designs triangular garden plots, each with two sides of lengths aa and bb and a fixed included angle of 35°35°. The table shows how the area changes as the product of the side lengths increases.

abab (cm²)20406080100120
Area (cm²)5.711.517.223.028.734.5
a
State the formula for the area of a triangle given two sides and their included angle. Hence calculate the constant of proportionality kk when the included angle is 35°35°. [1]
b
Analyse the data to show that area is directly proportional to abab, and determine the constant of proportionality from the data. [2]
c
The architect needs a plot with an area of at least 40 cm². Justify whether a plot with side lengths a=12a = 12 cm and b=11b = 11 cm is sufficient to meet this requirement. [1]
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8QuestionSolving ASA and AAS Triangle SituationsAssessment Practice
4 marks~6 minCriterion D
A surveyor needs to find the width of a river. She stands at point AA on one bank and sights a tree TT on the opposite bank, where the angle between the riverbank and ATAT is 90°90°. She walks 5050 m along the bank to point BB and measures the angle ABT=38°ABT = 38°. The riverbanks are straight and parallel.
a
Calculate the width of the river ATAT. Show all working. [2]
b
Each angle measurement has a possible error of ±0.5°\pm 0.5°. Calculate the range of possible widths of the river. [1]
c
The surveyor's manager requires a measurement accurate to within ±1\pm 1 m. Evaluate whether the surveyor should submit this result or take additional measurements, justifying your answer with reference to the angular error and at least one real-world limitation of the model. [1]
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9QuestionMultiple Solutions in Ambiguous SSA CasesAssessment Practice
2 marks~3 minCriterion C
A surveyor models a triangular plot of land ABCABC, where AB=8 cmAB = 8\ \text{cm}, AC=6 cmAC = 6\ \text{cm}, and ∠ABC=30°\angle ABC = 30°.
a
Calculate the perpendicular height hh from vertex BB to side ACAC. [1]
b
The surveyor claims only one triangular plot is possible. Justify whether this claim is correct, referring to the relationship between hh, ACAC, and ABAB. [1]
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10QuestionDrawing Diagrams from Bearings DescriptionsAssessment Practice
8 marks~12 minCriterion B
A surveyor maps four survey stations P, Q, R, and S using the following bearings:

P to Q: 030°030°
Q to R: 120°120°
R to S: 210°210°
S to P: 300°300°

All bearings are measured clockwise from north.
a
Construct a clearly labelled diagram showing points P, Q, R, S and the four bearing lines. Mark a north arrow at each station and label each bearing. [2]
b
Deduce the interior angle at each vertex of quadrilateral PQRS using the bearing data. Present your results as four labelled values. [2]
c
Analyse the relationship between consecutive bearings and the interior angle formed at their shared vertex. Hence advise the surveyor whether this quadrilateral is suitable to represent a rectangular building plot, justifying your answer with reference to both the angle properties and the closure of the survey loop. [4]
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11QuestionUnderstanding Bearings as Directional AnglesAssessment Practice
6 marks~9 minCriterion C
A port is located at the origin (0,0)(0, 0) and a lighthouse is at coordinates (3,4)(3, 4) km, where the positive xx-axis points east and the positive yy-axis points north. A student proposes the following model to calculate the bearing of the lighthouse from the port:

θ=tan⁡−1 ⁣(xy),B=90°−θ\theta = \tan^{-1}\!\left(\frac{x}{y}\right), \qquad B = 90° - \theta
a
Calculate the bearing BB predicted by the student's model. [2]
b
Construct the correct bearing using a clearly labelled diagram and trigonometric reasoning. [2]
c
The lighthouse is the destination of a vessel that must follow a bearing accurate to within ±2°\pm 2° to avoid shallow waters. Advise the navigator whether the student's model should be used for this voyage, identifying the source of any error in the model. [2]
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12QuestionSolving Bearings Questions Using TrigonometryAssessment Practice
4 marks~6 minCriterion D
A coastguard ship is positioned 45 m above sea level on a cliff edge. The navigator observes a lighthouse at sea level at an angle of depression of 12°12°.
a
Calculate the horizontal distance, dd, from the cliff edge to the lighthouse. [2]
b
A maritime safety regulation requires any vessel to maintain a horizontal clearance of at least 200 m from the lighthouse. Advise the navigator whether the ship satisfies this regulation, justifying your answer with reference to the margin of safety. [2]
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13QuestionDefining Sine Cosine and TangentAssessment Practice
8 marks~12 minCriterion D
A surveyor stands at the top of a cliff and measures the angle of depression to a boat at sea as θ=12°\theta = 12°. The horizontal distance from the base of the cliff to the boat is 150150 m. The surveyor models the cliff height using h=dtan⁡θh = d\tan\theta, where hh is the cliff height and dd is the horizontal distance.
a
Calculate the height of the cliff. Show all working. [3]
b
The angle measurement has a possible error of ±0.5°\pm 0.5°. Calculate the cliff height for θ=11.5°\theta = 11.5° and θ=12.5°\theta = 12.5°, and state the resulting range of possible heights. [2]
c
Advise the surveyor whether the model h=dtan⁡θh = d\tan\theta and a single angle reading are sufficient for reliable cliff-height measurement. In your advice, discuss how the angle measurement error affects the reliability of the result, identify at least two real-world factors that could reduce the model's accuracy, and justify your recommendation. [3]
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14QuestionUsing Trig Ratios to Find Missing AnglesAssessment Practice
12 marks~18 minCriterion B
A surveyor uses a drone to photograph a hillside. The drone records the horizontal distance (adjacent) and vertical rise (opposite) for three sight lines from a fixed base point.

Sight line 1opposite = 1 madjacent = 2 m
Sight line 2opposite = 2 madjacent = 3 m
Sight line 3opposite = 3 madjacent = 4 m


The angle of elevation θ\theta is measured from the horizontal.
a
Calculate θ1\theta_1, θ2\theta_2, and θ3\theta_3, the angles of elevation for sight lines 1, 2, and 3. [3]
b
Deduce the angle of elevation θ4\theta_4 for a fourth sight line where opposite = 4 m and adjacent = 5 m. Justify whether the angles of elevation are increasing at a constant rate. [4]
c
The surveyor requires an angle of elevation of exactly 45° to calibrate the camera. Analyse whether this calibration angle can ever be achieved using this sequence of sight lines, and evaluate what alternative measurement setup would be required. [5]
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15QuestionSolving Word Problems with Right-Angled TrianglesAssessment Practice
8 marks~12 minCriterion D
A surveyor stands 50 m horizontally from each of five structures and records the following data.

ObjectTreeBuildingFlagpoleCliffTower
Height (m)15.028.035.042.050.0
Angle of elevation (θ\theta)16.7°29.2°35.0°40.0°45.0°
a
Calculate tan⁡θ\tan\theta for each object, rounding to 3 decimal places. Present your results as labelled rows matching the format above. [2]
b
Analyse the relationship between height and tan⁡θ\tan\theta, using at least two numerical examples from your table as evidence. [2]
c
Deduce a general formula for height hh in terms of horizontal distance dd and angle of elevation θ\theta. Apply your formula to find the height of a lighthouse observed at an angle of elevation of 53.1° from a point 50 m away. [2]
d
A safety regulation requires a warning beacon on any structure taller than 65 m within 50 m of a public path. Evaluate whether the lighthouse requires a warning beacon, justifying your decision with reference to the precision of your calculation. [2]
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16QuestionCombining Pythagoras and Trig in 3D GeometryAssessment Practice
8 marks~12 minCriterion D
A designer models a square-based pyramid tent. The base is a square of side 6 m. Each triangular face has a slant height (the perpendicular distance from the midpoint of a base edge to the apex, measured along the face) of 8 m.

The diagram shows the tent with its key measurements labelled.
a
Show that the vertical height of the tent is h=55h = \sqrt{55} m. [2]
b
Calculate the total canvas area of the four triangular faces. Give your answer in m2^2 to 3 significant figures.

The designer claims the total canvas area needed is at least 95 m2^2. Justify whether the designer's claim is correct. [3]
c
In reality, the measured vertical height of a prototype tent is 7.2 m. Use the formula below to calculate the percentage error between the model's predicted height and the measured height. Give your answer to 3 significant figures.

percentage error=∣predicted−actual∣actual×100%\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%

Advise the designer whether the model is sufficiently reliable to use in production, justifying your advice with reference to your calculated percentage error. [3]
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17QuestionCalculating Diagonals and Angles in CuboidsAssessment Practice
10 marks~15 minCriterion A
A rectangular box has a base measuring 33 cm by 44 cm and a height of hh cm. The space diagonal of the box has length 1313 cm.

The diagram shows the box with vertices AA, BB, CC, DD on the base and A′A', B′B', C′C', D′D' directly above them, where ACAC is the base diagonal and AC′AC' is the space diagonal.
a
Show that h=12h = 12 cm. [3]
b
Find the angle between the space diagonal AC′AC' and the base diagonal ACAC. Give your answer correct to one decimal place. [3]
c
A second rectangular box is geometrically similar to the first, with a space diagonal of length 2626 cm. The manufacturer claims the second box has a total surface area of 384384 cm2^2. Justify whether this claim is correct. [4]
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18QuestionApplications in Architecture and PhysicsAssessment Practice
8 marks~12 minCriterion C
The diagram shows a vertical tower of height hh metres standing on horizontal ground. An observer at point P measures the angle of elevation θ\theta to the top of the tower from four different horizontal distances dd metres from the base of the tower.

The recorded data are shown below.

θ\theta (degrees): 15, 30, 45, 60

dd (metres): 37.32, 17.32, 10.00, 5.77
a
Show that d=htan⁡θd = \dfrac{h}{\tan\theta}, by referring to the right-angled triangle formed by the tower, the ground, and the line of sight. [2]
b
Using the row θ=45°\theta = 45° from the table, find the height hh of the tower. Hence find the horizontal distance dd when θ=20°\theta = 20°. Give both answers in metres to 3 significant figures. [3]
c
A student claims: "As the angle of elevation doubles, the horizontal distance halves."

Using at least two pairs of values from the table, justify whether this claim should be accepted or rejected. Your justification must refer to the formula d=htan⁡θd = \dfrac{h}{\tan\theta} to explain why the relationship between dd and θ\theta is or is not proportional in the way the student describes. [3]
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19QuestionCalculating Diagonals and Angles in CuboidsAssessment Practice
10 marks~15 minCriterion B
The table below shows three cuboids with integer edge lengths.

Base length aa (cm)358
Base width bb (cm)41215
Height hh (cm)51317


For each cuboid, the space diagonal connects one corner of the base to the diagonally opposite corner of the top face. The angle θ\theta is measured between the space diagonal and the base plane.
a
Find the base diagonal dd and the angle θ\theta for each cuboid. Give your answers for θ\theta to the nearest degree. [3]
b
Show that tan⁡θ=1\tan\theta = 1 for every cuboid in the table, and hence write a general formula for θ\theta in terms of aa, bb, and hh. [3]
c
A fourth cuboid has base length a=6a = 6 cm, base width b=6b = 6 cm, and space diagonal of length 10 cm. Justify whether θ\theta for this cuboid is greater than, equal to, or less than 45∘45^\circ, without using a calculator. [4]
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20QuestionChoosing Between Sine Rule and Cosine RuleAssessment Practice
6 marks~9 minCriterion D
A surveyor measures the distance across a lake. From point PP on the shore, she identifies two points QQ and RR on the opposite bank. She measures PQ=120PQ = 120 m, PR=150PR = 150 m, and angle QPR=65∘QPR = 65^\circ. The diagram shows triangle PQRPQR.
a
State the trigonometric rule that should be used to find QRQR, and identify the configuration of given information that requires this rule. [1]
b
Calculate the distance QRQR. Give your answer in metres to 3 significant figures. [2]
c
The surveyor later discovers that every angle she measured was 5∘5^\circ too large, so the true angle QPRQPR is 60∘60^\circ. The percentage error in QRQR is defined as

percentage error=∣QR65−QR60∣QR60×100%\text{percentage error} = \frac{|QR_{65} - QR_{60}|}{QR_{60}} \times 100\%

where QR65QR_{65} is your answer from part (b) and QR60QR_{60} is the distance calculated with the true angle. Calculate the percentage error to 1 decimal place, then advise the surveyor whether this measurement is fit for practical use in construction or land registration, justifying your advice with reference to the calculated values. [3]
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21QuestionConditions for Using the Sine RuleAssessment Practice
9 marks~14 minCriterion B
The diagram shows triangle ABCABC with AB=10AB = 10 cm, BC=aBC = a cm, and ∠BAC=30∘\angle BAC = 30^\circ.
a
Given that a=8a = 8, use the sine rule to find the exact value of sin⁡(∠ACB)\sin(\angle ACB). [2]
b
Show that when a=8a = 8, both possible values of ∠ACB\angle ACB produce valid triangles, and state the number of distinct triangles. [3]
c
Justify the range of values of aa for which exactly two distinct triangles exist, referring to the conditions on sin⁡(∠ACB)\sin(\angle ACB) and the relative sizes of aa and ABAB. [4]
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22QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
8 marks~12 minCriterion A
Triangle ABCABC has sides BC=aBC = a, AC=bAC = b and included angle CC, as shown in the diagram. A perpendicular from BB meets line ACAC at point HH, so that BH=hBH = h.
a
Show that the area of triangle ABCABC can be written as
Area=12absin⁡C,\text{Area} = \tfrac{1}{2}ab\sin C,
by first expressing hh in terms of aa and CC, then substituting into Area=12×base×height\text{Area} = \tfrac{1}{2} \times \text{base} \times \text{height}. Define all variables you use. [3]
b
The formula is applied to a triangle in which a=9a = 9 cm, b=7b = 7 cm and the area is 2727 cm2^2. Find the two possible values of angle CC, giving your answers in degrees to one decimal place. [3]
c
Justify which of the two values of CC found in part (b) gives the larger side ABAB, without calculating ABAB for either case. Your justification must cite a relevant trigonometric property. [2]
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23QuestionChoosing Between Sine Rule and Cosine RuleAssessment Practice
8 marks~12 minCriterion C
A surveyor is mapping two triangular plots of land. The diagram shows both triangles with their given measurements.

For triangle ABCABC: AB=85AB = 85 m, AC=110AC = 110 m, and ∠BAC=62∘\angle BAC = 62^\circ.

For triangle PQRPQR: PQ=95PQ = 95 m, ∠QPR=48∘\angle QPR = 48^\circ, and ∠PQR=73∘\angle PQR = 73^\circ.
a
Calculate the area of triangle ABCABC, giving your answer in square metres to 3 significant figures. [2]
b
Find the length of PRPR, giving your answer in metres to 3 significant figures. [3]
c
A colleague suggests using the sine rule to find an unknown side whenever two sides and a non-included angle are given. Critique this suggestion, referring to the conditions that create ambiguity and explaining how the given information in triangle PQRPQR avoids this problem. [3]
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24QuestionAmplitude Period and Phase ShiftAssessment Practice
8 marks~12 minCriterion B
A coastal engineer models ocean wave height using functions of the form y=asin⁡(bx)+dy = a\sin(bx) + d, where b>0b > 0, xx is time in seconds, and yy is height in metres. Three recorded wave patterns are:

y=2sin⁡(3x)+1y = 2\sin(3x) + 1: amplitude =2= 2, period =2π3= \dfrac{2\pi}{3}, vertical shift =1= 1

y=5sin⁡(2x)−4y = 5\sin(2x) - 4: amplitude =5= 5, period =π= \pi, vertical shift =−4= -4

y=12sin⁡(4x)+3y = \dfrac{1}{2}\sin(4x) + 3: amplitude =12= \dfrac{1}{2}, period =π2= \dfrac{\pi}{2}, vertical shift =3= 3
a
Deduce the amplitude of y=asin⁡(bx)+dy = a\sin(bx) + d in terms of aa. [2]
b
Deduce the period of y=asin⁡(bx)+dy = a\sin(bx) + d in terms of bb. [2]
c
A new wave is modelled by y=−4sin⁡ ⁣(12x)+6y = -4\sin\!\left(\dfrac{1}{2}x\right) + 6. Determine the amplitude, period, and vertical shift of this wave. [2]
d
The engineer states that a buoy rated to operate within 22 metres of a 66-metre mean water level can safely operate in this wave pattern. Advise the engineer whether this claim is correct, justifying your answer with reference to the wave's range of heights. [2]

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25QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice
3 marks~5 minCriterion C
A sound engineer models the pressure wave of a pure musical tone using y=sin⁡xy = \sin x, where xx represents time in degrees over one complete cycle (0°≤x≤360°0° \leq x \leq 360°).

The function passes through (0°,0)(0°, 0), (90°,1)(90°, 1), (180°,0)(180°, 0), (270°,−1)(270°, -1), and (360°,0)(360°, 0).
a
Calculate sin⁡45°\sin 45°, giving your answer in exact form. [1]
b
Deduce the values of sin⁡135°\sin 135° and sin⁡225°\sin 225°, using symmetry properties of the sine function. [1]
c
The engineer states: "The pressure wave is perfectly balanced — positive and negative peaks are equal in magnitude throughout the cycle." Using your results from (a) and (b), and the value sin⁡315°\sin 315°, justify whether this statement is correct. [1]
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26QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice
6 marks~9 minCriterion D
The tide height at a coastal station is modelled by

h(t)=3.2sin⁡(30t)∘+4.5h(t) = 3.2\sin(30t)^\circ + 4.5

where hh is the height in metres and tt is the time in hours after midnight. Observed heights are recorded below.

Time (h)024681012
Height (m)4.57.25.81.33.86.94.5
a
Calculate the predicted tide heights at t=2t = 2 h and t=8t = 8 h. [2]
b
Calculate the percentage error between the predicted and observed heights at t=8t = 8 h, using

percentage error=∣predicted−observed∣observed×100%.\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{\text{observed}} \times 100\%. [2]
c
A harbour authority requires model predictions to be within 15% of observed heights at all recorded times to approve the model for operational use. Advise the harbour authority whether this model should be approved, justifying your answer using your results. [2]
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27QuestionSin and Cos rules based problemsAssessment Practice
2 marks~3 minCriterion D
A surveying team needs to find the straight-line distance across a river to a landmark CC on the opposite bank. Direct measurement is impossible.

The team stands at two points, AA and BB, on the same bank, measuring a baseline AB=84 mAB = 84\ \text{m}. They record the angle to the landmark from each point: ∠CAB=71°\angle CAB = 71° and ∠CBA=63°\angle CBA = 63°.

Apply the Sine Rule to find the distance ACAC, then advise the team on whether a rope of length 110 m110\ \text{m} is sufficient to reach the landmark from point AA. [2]
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28QuestionSin and Cos rules based problemsAssessment Practice
8 marks~12 minCriterion B
A surveying team records two side lengths and a non-included angle for four separate plots of land. For each plot, the team must determine how many distinct triangular boundaries are geometrically possible before construction can begin.

Plot Aa=8a = 8 cmb=12b = 12 cmA=30°A = 30°
Plot Ba=10a = 10 cmb=8b = 8 cmA=40°A = 40°
Plot Ca=6a = 6 cmb=10b = 10 cmA=25°A = 25°
Plot Da=7a = 7 cmb=7b = 7 cmA=50°A = 50°


Here aa is the side opposite angle AA, and bb is the other given side.
a
Calculate all possible values of angle BB for each plot using the sine rule. State the number of distinct triangles possible in each case. [4]
b
Deduce a general rule, in terms of aa, bb, and AA, that predicts when exactly two distinct triangles are possible. [2]
c
A construction permit is issued only when a unique triangular boundary exists. Justify which plots receive a permit, using your rule from part (b) and the geometry of the sine ratio to explain why ambiguous cases must be rejected. [2]

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29QuestionSin and Cos rules based problemsAssessment Practice
8 marks~12 minCriterion D
Two search-and-rescue stations, A and B, are positioned 50 km apart along a straight coastline. Station A records the bearing of a distress signal as 30°30° from the coastline; station B records it as 50°50° from the coastline. Each bearing has an uncertainty of ±1°\pm 1° due to equipment limitations.
a
Show that the distance from station A to the signal is approximately 38.9 km, using the Law of Sines with exact bearings of 30°30° and 50°50°. [2]
b
Deduce the maximum possible range of distances from station A to the signal when the worst-case bearing errors of ±1°\pm 1° are applied simultaneously at both stations. [2]
c
A rescue coordinator must decide whether two-station triangulation is reliable enough to direct a rescue vessel in rough seas. Advise the coordinator whether this method should be used alone or combined with supplementary positioning, using your results from parts (a) and (b) and at least two real-world limitations of the model. [4]
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30QuestionProblem Solving with Non-Right TrianglesAssessment Practice
6 marks~9 minCriterion B
A triangular solar panel frame is being designed with two fixed struts of length 10 cm and 14 cm. The included angle CC between the struts can be adjusted during assembly.
a
Using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C, calculate the area of the triangle for each angle below. [2]

Angle CC (degrees): 20, 40, 60, 80, 100

Area (cm²): ___, ___, ___, ___, ___
b
Describe the pattern in the areas as CC increases from 20° to 100°, and deduce the angle that produces the maximum area. [2]
c
Justify the engineer's claim that C=90°C = 90° produces the greatest area, using the behaviour of sin⁡C\sin C, and advise the assembly team on how to set the struts and what happens to the enclosed area if they deviate from this angle. [2]

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31QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
4 marks~6 minCriterion C
A landscape architect is designing a triangular garden bed with two fixed side lengths a=9.2a = 9.2 cm and b=12.5b = 12.5 cm (on a scale drawing). The graph below shows how the area of the triangle varies with the included angle CC, using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C.
a
Show that the area formula simplifies to Area=57.5sin⁡C\text{Area} = 57.5\sin C. [1]
b
Explain how the shape of the graph shows that the maximum area occurs at C=90°C = 90°. [2]
c
The architect claims a right-angled triangular bed gives the most efficient use of the fixed side lengths. Calculate the maximum area correct to 3 significant figures, and advise the architect whether this right-angle design should be adopted to maximise the garden bed area. [1]
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32QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
4 marks~6 minCriterion D
A surveyor is mapping a triangular plot of land, triangle PQRPQR, where PQ=10PQ = 10 m, PR=14PR = 14 m, and angle Q=40°Q = 40°. The angle at QQ is not the included angle between sides PQPQ and PRPR.

Local planning regulations require any triangular plot submitted for residential development to have an area of at least 6060 m².
a
Use the sine rule to calculate angle RR. Give your answer to one decimal place. [1]
b
Calculate the area of triangle PQRPQR using the formula Area=12absin⁡C\text{Area} = \dfrac{1}{2}ab\sin C. Give your answer to two decimal places. [1]
c
Advise the surveyor whether this plot should be submitted for residential development, referring to the area requirement and the reliability of the result. [2]
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